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On geodesic exponential maps of the Virasoro group. (English) Zbl 1121.35111
Summary: We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ (k) (k 0) on the Virasoro group Vir and show that for k2, but not for k=0,1, each of them defines a smooth Fréchet chart of the unital element e Vir. In particular, the geodesic exponential map corresponding to the Korteweg-de Vries (KdV) equation (k=0) is not a local diffeomorphism near the origin.
MSC:
35Q35PDEs in connection with fluid mechanics
37K30Relations of infinite-dimensional systems with algebraic structures
37K25Relations of infinite-dimensional systems with differential geometry
58B25Group structures and generalizations on infinite-dimensional manifolds
References:
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